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HARMONIC FUNCTIONS ON MULTIPLICATIVE GRAPHS AND INTERPOLATION POLYNOMIALS Alexei Borodin and Grigori Olshanski Department of Mathematics The University of Pennsylvania Philadelphia, PA 19104-6395, U.S.A. borodine@math.upenn.edu Dobrushin Mathematics Laboratory, Institute for Problems of Information Transmission, Bolshoy Karetny 19, 101447 Moscow GSP-4, RUSSIA. olsh@iitp.ru, olsh@glasnet.ru Submitted: November 22, 1999; Accepted: May 15, 2000. We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multi- variate interpolation polynomials associated with Schur’s S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type inte- grals. Contents §0. Introduction §1. The general formalism §2. The Young graph §3. The Jack graph §4. The Kingman graph §5. The Schur graph §6. Finite–dimensional specializations 6.1. Truncated Young branching 6.2. Γ–shaped Young branching 1991 Mathematics Subject Classification. Primary 05E10, Secondary 31C20, 60C05. One of the authors (G.O.) was supported by the Russian Foundation for Basic Research under grant 98–01–00303. Typeset by A M S-T E X 1 7 2 6.3. Truncated Kingman branching 6.4. Truncated Schur branching §7. Appendix References §0. Introduction Let Y denote the lattice of Young diagrams ordered by inclusion. For µ, λ ∈ Y, we write λ  µ if λ covers µ, i.e., λ differs from µ by adding a box. We consider Y as a graph whose vertices are arbitrary Young diagrams µ and the edges are couples (µ, λ) such that λ  µ. We shall call Y the Young graph. A function ϕ(µ) is called a harmonic function on the Young graph [VK] if it satisfies the condition ϕ(µ)=  λ: λµ ϕ(λ), ∀µ ∈ Y. (0.1) We are interested in nonnegative harmonic functions ϕ normalized at the empty diagram: ϕ(∅) = 1. Such functions form a convex set denoted as H + 1 (Y). The functions ϕ ∈H + 1 (Y) have an important representation–theoretic meaning: they are in a natural bijective correspondence with central, positive definite, nor- malized functions on the infinite symmetric group S(∞), see [VK], [KV2]. Thoma’s description of characters on S(∞) means that the extreme points of H + 1 (Y) form an infinite–dimensional simplex Ω (called the Thoma simplex), see [T], [VK], [KV2], [W]. For general elements ϕ ∈H + 1 (Y), there is a (unique) Poisson–type integral representation, ϕ(λ)=  Ω K(λ, ω) P(dω), ∀λ ∈ Y, (0.2) where P is a probability measure on Ω (the ‘boundary measure’ for ϕ)andK(λ, ω) is a positive function on Y × Ω (the ‘Poisson kernel’ or ‘Martin kernel’ for Y), see [KOO]. Note that any probability measure P on Ω gives rise to an element of H + 1 (Y); in particular, the extreme ϕ’s are exactly the functions K(·,ω) corresponding to Dirac measures on Ω. This abstract result shows how large H + 1 (Y) is but it does not explain how to construct explicitly nonextreme functions ϕ or what nonextreme ϕ’s could be interesting for applications. Concrete examples of nonextreme functions ϕ first emerged in [KOV] in connec- tion with a problem of harmonic analysis on the infinite symmetric group S(∞). These functions, denoted as ϕ zz  , depend on two parameters, and the correspond- ing ‘boundary measures’ P zz  govern the spectral decomposition of certain natural unitary representations. 1 1 The measures P zz  are very interesting objects. They are studied in detail in our papers [P.I] – [P.V], [BO1], [BO2]. 7 3 The explicit expression of the functions ϕ zz  (see formula (2.4) below) has an interesting combinatorial structure which raises a number of questions. For instance, one can ask whether there exist similar families of harmonic functions for other graphs. The answer is affirmative: [B1], [Ke5]. 2 The paper [B1] concerns the graph S of shifted Young diagrams which is related to projective representations of the symmetric groups. The paper [Ke5] contains a generalization in another direction: a deformation of the family {ϕ zz  }, which is consistent with a deformation of the basic equation (0.1): ϕ(µ)=  λ: λµ κ θ (µ, λ)ϕ(λ), ∀µ. (0.3) Here θ>0 is the deformation parameter and κ θ (µ, λ) > 0arethecoefficients that arise in (the simplest case of) Pieri’s rule for Jack symmetric functions with parameter θ. The initial situation corresponds to the particular value θ =1,when Jack symmetric functions coincide with Schur’s S–functions. Note that in the limit as θ → 0, the harmonicity condition (0.3) essentially coin- cides with the relation which defines partition structures in the sense of Kingman [Ki1], [Ki2], while the two-parameter family of harmonic functions constructed in [Ke5] degenerates to the famous Ewens partition structures [Ew] and its general- ization due to Pitman, see [Pi], [PY], [Ke4]. In the present paper, we propose a simple combinatorial construction, which allows us to get, in a unified way, all these concrete examples of harmonic functions as well as some new ones. In the new examples, the ‘boundary measures’ P are supported by finite–dimensional simplices, and the Poisson integral representation leads to certain Selberg–type integrals. 3 Our construction relies on the so–called shifted (or factorial) versions of Schur’s S and P functions and of Jack symmetric functions. These new combinatorial functions arise in different topics, see, e.g., [S], [KS], [OO], [OO2], [Ok1], [Ok2]. They are also called interpolation polynomials, because they give solutions to certain multivariate interpolation problems. The paper is organized as follows. In §1, we expose the general formalism. In §2, it is applied to the Young graph to derive the family {ϕ zz  }.In§§3–5, we apply it to the Young graph with Jack edge multiplicities J(θ), next to the Kingman graph K, and then to the Schur graph S; the arguments are quite similar. Section 6 is devoted to constructing harmonic functions of a different sort — those with finite– dimensional ‘boundary measures’; here we also evaluate Selberg–type integrals. The final §7 is an appendix on the Poisson integral representation. 2 Another question, a characterization of the functions of type ϕ zz  , was examined in [B1], [Ro]. 3 A connection between Poisson integral representation of type (0.2) and Selberg integrals was first exploited in [Ke3]. 7 4 §1. The general formalism In this section, we deal with an abstract graph G satisfying certain conditions listed below. In the next sections, concrete examples of G will be considered. Our assumptions and conventions concerning G are as follows: • To simplify the notation, we identify the graph with its set of vertices. • The vertices are partitioned into levels, G = G 0  G 1  G 2  , so that the endpoints of any edge lie on consecutive levels. That is, G is a graded graph. • The level of a vertex µ is denoted as |µ|. If two vertices µ, λ form an edge, |λ| = |µ| + 1, then we write λ  µ or µ  λ. • All the levels G n are finite. • The lowest level G 0 consists of a single vertex denoted as ∅. • For any vertex µ there exists at least one vertex λ  µ and for any vertex λ = ∅ there exists at least one vertex µ  λ. This implies that the graph is connected. (Our main example is the Young graph, see §2. ) • Finally, assume that we are given an edge multiplicity function which assigns to any edge µ  λ a strictly positive number κ(µ, λ) — its formal multiplicity. It should be emphasized that these numbers are not necessarily integers. (For the Young graph, all the formal multiplicities are equal to 1; graphs with nontrivial multiplicities are considered in §3and§4.) A complex function ϕ(µ)onG is called a harmonic function on the graph G if it satisfies the relation ϕ(µ)=  λ: λµ κ(µ, λ)ϕ(λ) (1.1) for any vertex µ (the sum in the right–hand side is finite, because all the levels are finite). Let H(G) denote the space of all harmonic functions endowed with the topology of pointwise convergence. Let H + (G) be the subset of nonnegative harmonic functions and H + 1 (G) be the subset of the functions ϕ ∈H + (G)withthe normalization ϕ(∅)=1. Clearly, H + 1 (G) is a convex subset of H(G). Moreover, it is a compact measurable space (here we again employ the finiteness assumption). We shall use some well– known general theorems about convex compact measurable sets which can be found, e.g., in [Ph]. Let Ω(G) denote the set of extreme points in H + 1 (G). This is a set of type G δ , hence, a Borel measurable set. Given ω ∈ Ω(G),letusdenotebyK( · ,ω) the corresponding extreme harmonic function on G. Note that K(µ, · )isaBorel measurable function on Ω(G) for any fixed µ ∈ G. Theorem 1.1. For each element ϕ ∈H + 1 (G) there exists a unique probability mea- sure P on Ω(G) such that ϕ(µ)=  Ω(G) K(µ, ω)P (dω), ∀µ ∈ G. (1.2) 7 5 Proof. See §7.  We call (1.2) the Poisson integral representation of the function ϕ. To any path τ going from a vertex µ toavertexλ with |λ| > |µ|, τ =(µ = λ 0  λ 1 ···λ k = λ),k= |λ|−|µ|, we assign its weight w(τ)= k  i=1 κ(λ i−1 ,λ i ) andthenset dim G (µ, λ)=  τ w(τ), (1.3) summed over all paths from µ to λ. We extend this definition to all couples (µ, λ) by agreeing that dim G (µ, µ) = 1 and dim(µ, λ)=0ifµ = λ are such that there is no path from µ to λ. Next, we set dim G λ =dim G (∅,λ). In all examples of the graphs G considered in the present paper one can embed (the vertices of) G into Ω(G) in such a way that any point ω ∈ Ω(G)canbe approximated by a sequence of vertices {λ(n) ∈ G n } n=1,2, , and for any such sequence K(µ, ω) = lim n→∞ dim G (µ, λ(n)) dim G λ(n) . Given a function ϕ ∈H + 1 (G), we set for each n M n (λ)=dim G λ · ϕ(λ),λ∈ G n . (1.4) Using the harmonicity relation (1.1) and induction on n one readily verifies that  λ∈G n M n (λ) = 1. Thus, each M n is a probability distribution on G n . For all examples of the graphs G considered in this paper one can transfer the measure M n to Ω(G) via the embedding G → Ω(G) mentioned above. Then the measure P appearing in the integral representation (1.2) is the weak limit of the measures M n as n →∞. We say that (G, κ( · , · )) is a multiplicative graph [KV1], [KV2], if the following conditions are satisfied. First, the 1st floor G 1 consists of a single vertex denoted by the symbol “(1)”. Next, there exists a graded commutative unital algebra A over R, A = A 0 + A 1 + , and a homogeneous basis {P µ } in A indexed by the vertices µ ∈ G, such that P ∅ =1anddegP µ = |µ|. Finally, for any µ, P µ P (1) =  λ: λµ κ(µ, λ)P λ . (1.5) Note that this implies κ(∅, (1)) = 1. (All the graphs considered in the present paper are multiplicative. For instance, in the case of the Young graph, the algebra 7 6 A is the algebra of symmetric functions and the basis {P µ } is formed by the Schur functions.) Iterating the relation (1.5) we get the expansion P n (1) =  λ∈G n dim G λ ·P λ , (1.6) which is a useful tool for computing the dimensions dim G λ. More generally, given µ ∈ G m and n>m, P µ P n−m (1) =  λ∈G n dim G (µ, λ) ·P λ . (1.7) Theorem 1.2 [KV1]. Let G be a multiplicative graph and let A be the corresponding algebra. Given ϕ ∈H + 1 (G),letπ : A → C be the linear functional sending each P µ to ϕ(µ).Thenϕ is extreme if and only if π is multiplicative. Note that a linear functional π : A → C corresponds to a function ϕ ∈H + 1 (G)if and only if π(1) = 1, π(P µ ) ≥ 0 for any µ,andπ factors through A/(P (1) − 1)A. Now we shall explain our method of producing harmonic functions. Assume A ∗ is a commutative algebra 4 , {P ∗ µ } is a family of elements in A ∗ indexed by the vertices µ ∈ G, P ∗ ∅ = 1. We assume that these data obey the following condition which is a generalization of (1.5): P ∗ µ P ∗ (1) = a n P ∗ µ +  λ: λµ κ(µ, λ)P ∗ λ ,n= |µ|, (1.8) for any µ,wherea 0 =0,a 1 ,a 2 , is a sequence of numbers. Proposition 1.3. Under the above assumptions, let π : A ∗ → C be a multiplicative linear functional, and let s = π(P ∗ (1) ),t= −s = −π(P ∗ (1) ). (1.9) Assume that s =0,a 1 ,a 2 , , i.e.,t=0, −a 1 , −a 2 , . (1.10) Then the function ϕ(µ)= π(P ∗ µ ) s(s − a 1 ) (s − a n−1 ) = (−1) n π(P ∗ µ ) t(t + a 1 ) (t + a n−1 ) ,n= |µ|, (1.11) is harmonic on G. We agree that the denominator in (1.11) equals 1 for µ = ∅,sothatϕ(∅)=1. 4 The superscript ∗ does not mean the passage to a dual space. 7 7 Proof. Applying π to the relation (1.8) we get π(P ∗ µ )(s − a n )=  λ: λµ κ(µ, λ)π(P ∗ λ ). Dividing the both sides by s(s − a 1 ) (s − a n ) (which is possible thanks to (1.10)) we get exactly the harmonicity relation (1.1) for ϕ.  A trivial example is A ∗ = A, P ∗ µ = P µ , a n ≡ 0. Then, by Theorem 1.2, ϕ is extreme provided that it is nonnegative. As we aim to construct interesting examples of nonextreme harmonic functions, we shall deal either with an algebra A ∗ distinct from A or, for A ∗ = A, with a family {P ∗ µ } distinct from {P µ }. In all the examples below, A ∗ is a filtered algebra such that the associated graded algebra gr A ∗ is canonically isomorphic to A. Thus, with any element of A ∗ of degree ≤ n one can associate its highest term which is a homogeneous element of A of degree n. In our examples, the highest term of P ∗ µ coincides with P µ . Furthermore, the algebra A ∗ can be interpreted, in a certain natural way, as an algebra of functions on the vertices of G. Thus, for any f ∈ A ∗ and λ ∈ G,thevaluef(λ) is well–defined. It turns out that the elements P ∗ µ can be characterized by the following Interpolation Property. Given µ ∈ G, µ = ∅, P ∗ µ is the only (up to a scalar factor) element of degree |µ| such that P ∗ µ (λ) = 0 for any λ = µ with |λ|≤|µ|. The fact that the highest term of an element P ∗ µ defined in this way turns out to be proportional to P µ seems to be rather surprising. We normalize P ∗ µ in such a way that its highest term is exactly equal to P µ . Next, it turns out that P ∗ (1) (µ)=|µ|. Then a simple formal argument shows that (1.8) holds with a n = n for any n =0, 1, .Moreover, dim G (µ, λ) dim G λ = P ∗ µ (λ) N(N − 1) (N − n +1) ,µ∈ G n ,λ∈ G N ,n≤ N. (1.12) The argument is due to Okounkov [Ok1]; it is also reproduced in [OO]. From now on we shall assume that a n = n. Then the denominator in the right– hand side of (1.11) will be equal to (t) n = t(t +1)···(t + n − 1), and (1.11) will take the form ϕ(µ)= (−1) n π(P ∗ µ ) (t) n ,t= −π(P ∗ (1) ),n= |µ|. (1.13) Similarly, the formula (1.12) can be rewritten as follows dim G (µ, λ) dim G λ = (−1) n P ∗ µ (λ) (−N) n ,µ∈ G n ,λ∈ G N ,n≤ N. (1.14) 7 8 Note that, for any fixed λ, the left–hand side of (1.14) satisfies the harmonicity relation (1.1) provided that n<N: this easily follows from the very definition of the dimension function (for n>N the denominator in the right–hand side vanishes). On the other hand, the expression in the right–hand side of (1.14) is a particular case of that in the right–hand side of (1.13): here π is the evaluation functional π λ : f → f (λ)andt = −N. This makes it possible to interpret the construction of Proposition 1.3 as follows: we extrapolate the relation (1.14) from the points λ ∈ G, which we identify with the corresponding evaluation functionals π λ ,to abstract multiplicative functionals. A function ϕ ∈H + 1 (G) will be called nondegenerate if ϕ(µ) =0forallµ ∈ G; otherwise it will be called degenerate. §2. The Young graph The fundamental example of a graded graph G is the Young graph Y [VK], [KV2]. By definition, the vertices of Y are the Young diagrams including the empty diagram ∅,then-th floor Y n consists of the diagrams with n boxes, and µ  λ means that λ is obtained from µ by adding a single box. The numbers κ(µ, λ) are all equal to 1. In this section the symbols µ, λ are used to denote Young diagrams. The graph Y is multiplicative in the sense of the definition given in §1: here the algebra A is the algebra Λ of symmetric functions, the basis elements P µ are the Schur functions s µ , and the relation (1.5) turns into a special case of the Pieri rule for the Schur functions, s µ s (1) =  λ: λµ s λ , (2.1) which is equivalent (under the characteristic map, see [M, I.7]) to the Young branch- ing rule for irreducible characters of symmetric groups. For the Young graph, the expansion (1.6) takes the form s n (1) =  λ: |λ|=n dim λ · s λ , (2.2) where dim λ =dim Y λ is the number of standard Young tableaux of shape λ. Let b =(i, j)beaboxofµ;herei, j are the row number and the column number of b. Recall the definition of the content,thearm–length and the leg–length of b: c(b)=j − i, a(b)=µ i − j, l(b)=µ  j − i, (2.3) where µ  is the transposed diagram. Theorem 2.1. Let z, z  be arbitrary complex numbers and t = zz  . Assume that t =0, −1, −2, . Then the following expression is a harmonic function on the Young graph: ϕ zz  (µ)= 1 (t) n  b∈µ (z + c(b))(z  + c(b)) a(b)+l(b)+1 ,n= |µ|. (2.4) 7 9 The harmonic functions (2.4) fit into the general scheme of Proposition 1.3 with the algebra A ∗ and the family {P ∗ µ } as specified below. The first claim of the theorem (harmonicity of ϕ zz  ) follows from the computation of a spherical function in [KOV]. Various direct combinatorial proofs for this claim were given by Kerov, Postnikov, and Borodin. Kerov’s approach is explained in [Ke5]; actually, in that paper a more general result is obtained, see Theorem 3.1 below. Postnikov’s argument was not published. Borodin’s argument is, perhaps, the most direct and elementary; it was given in the appendix to [P.I]; actually, the present paper originated from our discussion of that argument. For the proof we need some preparations. First, we specify the algebra A ∗ . Denote by Λ ∗ (n) the subalgebra in C[x 1 , ,x n ] formed by the polynomials which are symmetric in ‘shifted’ variables x  j = x j − j, j =1, ,n. Define the projection map Λ θ (n) → Λ θ (n − 1) as the specialization x n = 0 and note that this projection preserves the filtration defined by ordinary degree of polynomials. Now we take the projective limit of Λ ∗ (n)’s in the category of filtered algebras as n →∞. The result is a filtered algebra which is called the algebra of shifted symmetric functions and denoted by Λ ∗ . The algebra Λ ∗ will be taken as the algebra A ∗ . As the elements P ∗ µ we shall take the shifted Schur functions s ∗ µ as defined in [OO]. By the definition of Λ ∗ ,eachelementf ∈ Λ ∗ can be evaluated at any sequence x =(x 1 ,x 2 , ) with finitely many nonzero terms. In particular, we can evaluate shifted symmetric functions at any λ =(λ 1 ,λ 2 , ) ∈ Y, which allows one to interpret Λ ∗ as a certain algebra of functions on the Young diagrams. This point of view was developed in [KO]. The shifted Schur functions s ∗ µ possess the Interpolation Property of §1, see [Ok1], [OO]. For the one–row shifted Schur functions there is a special notation: h ∗ m = s ∗ (m) . A useful tool is the following generating series for the h ∗ functions: H ∗ (u)=1+ ∞  m=1 h ∗ m u(u − 1) (u − m +1) . (2.5) Here u is a formal indeterminate and the series is viewed as an element of Λ ∗ [[ 1 u ]]. Since the elements h ∗ m are algebraically independent generators of Λ ∗ , a multiplica- tive functional π :Λ ∗ → C can be uniquely defined by assigning to H ∗ (u)an arbitrary formal power series in 1 u with constant term 1. We shall use this fact below. Note a useful formula H ∗ (u)(x 1 ,x 2 , )= ∞  i=1 u + i u + i − x i , (2.6) 7 10 see [OO, Theorem 12.1]. Here, by definition, H ∗ (u)(x 1 ,x 2 , )=1+ ∞  m=1 h ∗ m (x 1 ,x 2 , ) u(u − 1) (u − m +1) . (2.7) The equality (2.6) can be understood as follows. We assume that only finitely many of x i ’s are distinct from zero. Then the left–hand side, which is the series (2.7), converges in a left half–plane u<const  0 and equals the right–hand side of (2.6). For an element f of Λ or Λ ∗ , we shall abbreviate f(x 1 , ,x k )=f(x 1 , ,x k , 0, 0, ). Recall the combinatorial formula for the Schur functions: s µ (x 1 , ,x k )=  T  b∈µ x T (b) , (2.8) where T ranges over the set of Young tableaux of shape µ with entries in {1, ,k}, see [M, I.5]. It will be convenient for us to employ here the reverse tableaux (i.e., the entries T(b) decrease from left to right along the rows and down the columns). Since s µ is symmetric, (2.8) also holds if the sum in the right–hand side is taken over all reverse tableaux of shape µ with entries in {1, ,k}. We shall need a similar formula for the shifted Schur functions: s ∗ µ (x 1 , ,x k )=  T  b∈µ (x T (b) − c(b)) , (2.9) where T ranges over reverse tableaux of shape µ with entries in {1, ,k}, see [OO, Theorem 11.1]. Proposition 2.2. Let k =1, 2, and z  ∈ C. The following specialization for- mula holds s ∗ µ (−z  , ,−z     k )=(−1) n  b∈µ (k + c(b))(z  + c(b)) a(b)+l(b)+1 ,n= |µ|. (2.10) Proof. Compare the combinatorial formulas (2.8) and (2.9). If x 1 = ···= x k = −z  then the product in (2.9) does not depend on T and is equal to  b∈µ (−z  − c(b)) = (−1) n  b∈µ (z  + c(b)). [...]... πk(1−k)/2 = evaluation at the staircase diagram (k, k − 1, , 1) (5.12) This claim is an analog of Propositions 2.2, 3.2, and formula (4.5) It does not seem to have appeared in the literature before, so we give here a sketch of the proof We shall consecutively check (5.12) on the one–row P ∗ functions, next on the two–row P ∗ functions, and finally on arbitrary P ∗ functions First, consider the generating... summation formula, cf (2.12) When t1 , t2 are as in (5.11), this coincides with (5.13) Thus, we have checked (5.12) on the one–row functions Next, we employ recurrence relations which make it possible to express the two– row functions through the one–row ones It is convenient to extend the definition ∗ of two–row functions P(p,q) with p > q ≥ 1 to a larger set of indices by adopting the ∗ ∗ ∗ convention... the two–row functions through the one–row functions A direct computation shows that the relations (5.15), (5.16) remain valid if we apply πt to each P ∗ function involved This means that (5.12) holds on the two–row functions Finally, to handle arbitrary P ∗ functions we employ the following relation proved in [I2]: ∗ ∗ Pµ = Pf P(µi ,µj ) (5.17) 1≤i,j≤ (µ)+ε Here the symbol Pf means Pfaffian and ε equals... , xl+2 , to zero In any such specialization all functions s∗ with the length (number of nonzero µ parts) of µ greater than l vanish, see [OO] This implies that the harmonic function on Y afforded by Proposition 1.3 vanishes on all Young diagrams with more than l rows Thus, one can consider such a function as a harmonic function on the subgraph Y(l) of Y consisting of all Young diagrams with length... mentioned in §2, the set H1 (Y) = H1 (J(1)) has a representation theoretic meaning There is one more value of θ, namely θ = 1/2, when harmonic functions on the Jack graph can be related to representations We shall briefly explain this connection Let G be the group of finite permutations of the set {±1, ±2, } and K be its subgroup consisting of the permutations which commute with the involution i →... ; here the second arrow S(n) → Kn assigns to a permutation its cycle structure which is identified with a partition, and (Mn ) is the partition structure corresponding to ϕ Let us denote by Mt,α the measures corresponding to the harmonic functions ϕt,α , where the parameters satisfy the conditions of Proposition 4.2 The measure M1,0 is invariant with respect to S(∞)×S(∞) and it is the only probability... finite symmetric groups In other words, S(∞) consists of the finite permutations of the set {1, 2, } There is a natural bijective correspondence ϕ ↔ χ between functions ϕ ∈ H(Y) and central functions χ on S(∞) Specifically, given χ, we define the values of ϕ on Yn from the expansion of the central function χ ↓ S(n) on the group S(n) into a linear combination of the irreducible characters χλ , χ ↓ S(n)... definition of the Jack functions, their expansion in the monomial functions has the form Pµ = mµ + lower terms relative to the dominance order on partitions [M, VI, (10.13)] It is well known that in the limit θ → 0 the coefficients of all the lower terms vanish In this sense, the Jack functions Pµ degenerate to the monomial functions mµ as θ → 0 This implies, in particular, the limit relation κ0 (µ, λ) = lim... (5.19) §6 Finite–dimensional specializations In the previous four sections we described four different examples of graphs which fit into the general scheme introduced in §1 For each of these graphs we produced a nontrivial family of specializations of the corresponding algebras A∗ which defined, according to Proposition 1.3, a certain family of (nonnegative) harmonic functions on the graph Every such... for the Kingman graph the theory is connected with Poisson processes Our goal in this section is to construct ‘simpler’ families of harmonic functions for Young, Kingman, and Schur graphs The word ‘simpler’ means that the corresponding measures on Ω(G) will be supported by finite–dimensional subspaces These measures will be explicitly computed The corresponding Poisson integrals (which arise due to Theorem . harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multi- variate. applications. Concrete examples of nonextreme functions ϕ first emerged in [KOV] in connec- tion with a problem of harmonic analysis on the infinite symmetric group S(∞). These functions, denoted. multi- variate interpolation polynomials associated with Schur’s S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type inte- grals. Contents §0. Introduction §1.